The case of the biased quenched trap model in two dimensions with diverging mean dwell times

TL;DR

This paper analyzes the biased quenched trap model in two dimensions with diverging mean dwell times, deriving explicit probability distributions, moments, and non-linear responses in time and force, supported by numerical simulations.

Contribution

It provides an explicit disorder-averaged probability density function and moments for the 2D biased quenched trap model with diverging dwell times, including non-linear response behaviors.

Findings

Explicit probability density function derived for large times.

First and second moments calculated, with a general formula for the $$-th moment.

Non-linear response in time and external force observed and characterized.

Abstract

We investigate the biased quenched trap model on top of a two-dimensional lattice in the case of diverging expected dwell times. By utilizing the double-subordination approach and calculating the return probability in $2$d, we explicitly obtain the disorder averaged probability density function of the particle's position as a function of time (for any given bias) in the limit of large times ($t \rightarrow \infty$). The first and second moments are calculated, and a formula for a general $\mu$-th moment is found. The behavior of the first moment, i.e. $\langle x(t)\rangle$, presents non-linear response both in time and in the applied external force $F_0$. While the non-linearity in time occurs for any measurement time $t$, the non-linearity in $F_0$ is expected only when $t\gtrsim \big(F_0 \left| \ln (F_0)\right|\big)^{-2 / \alpha}$ where $\alpha=T/T_g$, for temperatures $T<T_g$. We support our analytic results by comparison to numerical simulations.